Defect-gradient-induced Rashba effect in van der Waals PtSe2 layers

Defect engineering is one of the key technologies in materials science, enriching the modern semiconductor industry and providing good test-beds for solid-state physics. While homogenous doping prevails in conventional defect engineering, various artificial defect distributions have been predicted to induce desired physical properties in host materials, especially associated with symmetry breakings. Here, we show layer-by-layer defect-gradients in two-dimensional PtSe2 films developed by selective plasma treatments, which break spatial inversion symmetry and give rise to the Rashba effect. Scanning transmission electron microscopy analyses reveal that Se vacancies extend down to 7 nm from the surface and Se/Pt ratio exhibits linear variation along the layers. The Rashba effect induced by broken inversion symmetry is demonstrated through the observations of nonreciprocal transport behaviors and first-principles density functional theory calculations. Our methodology paves the way for functional defect engineering that entangles spin and momentum of itinerant electrons for emerging electronic applications.

This manuscript presents an investigation of the potential for engineering a Rashba field within a 2D material using a vertical gradient in the density of structural defects. The material of choice is PtSe2, and the authors verify the creation of the structural defects with a combination of electron microscopy and spectroscopy. The impact of the potential Rashba field is measured using longitudinal magneto-transport, and the authors demonstrate the onset of an asymmetry in the MR as a function of current direction in samples with defect gradients that is not present in pristine samples. A theoretical basis for this effect is presented using a combination of referencing to prior literature and the presentation of DFT calculations of pristine and defected material.
The material characterization is extremely thorough and the authors demonstrate the creation of the defect gradient that they are attempting to engineer in a very compelling fashion. Similarly, the presence of the asymmetry in magnetoresistance is also clear and indexing to control samples provides a compelling argument that it is related to the defect incorporation. The theoretical foundations for assigning this MR to a Rashba field was less clear to me, and I am unsure if this is simply an issue of clarity in the manuscript or if there are more fundamental issues. As a result, while I believe there is the potential for impactful science here I would like to see a revised manuscript that addresses the following points before advocating for publication.
1. There is some confusion regarding the orientation of the respective fields. First, it is my understanding that the native spin orbit coupling (SOC) in PtSe2 is out of plane (in the z direction according to the authors convention). Since this is the same direction as the Rashba field proposed as arising from the defects I would appreciate some discussion of how these two effects are distinct and/or can be disentangled. Second, the labeling in the SEM image of their device in Fig 1  is confusing. The image is zoomed in so that only a series of stripes is visible, which I take to be the transport channels of the PtSe2. Under this convention the authors appear to be proposing a transverse current while their measurements are longitudinal. I think a much more careful description of the sample geometry and the symmetry of the relevant effects is warranted.
2. The role of the DFT in validating the proposed mechanism is not clear to me. The two plots presented are DFT with and without spin orbit coupling, but as noted above PtSe2 has native SOC that must be taken into account. Are the authors explicitly taking into account different levels of defect incorporation into account in their calculations? If so, there is a high computational cost for the more heavily defected layers and I would be interested to hear more about how they manage this calculation. If not, then the applicability of this DFT work is not clear. How does it support the hypothesis that the defect gradient is giving rise to a spin-orbit field with the appropriate symmetry to generate the experimental results? If the authors are just manually tuning the strength of SOC to mimic what they expect to happen with the different defect densities then this needs to be clearly stated and the implications of this approach need to be more thoroughly discussed.
In summary, this is some very thorough experimental work with a clear signature of current asymmetry, but the discussion of possible mechanisms does not allow for a careful evaluation of whether the author's proposed model is credible. In the absence of such a theoretical understanding I think the impact of the work may be somewhat more narrow than is appropriate for publication in Nature Communications, so this question needs to be resolved prior to publication.

Our Response:
We greatly appreciate the reviewer's important comments as well as positive remarks on our manuscript. We believe that our approach through a defect-gradient would be one of the novel methods to induce inversion symmetry breaking, in particular, in PtSe2 as well as 2D layered materials that represent unique and superior properties throughout various scientific realms. The following is detailed discussions and corrections we made in response to the reviewer's comments to improve the quality of this manuscript.

#1. Reviewer's Comments:
The authors achieve a defect gradient with a dept of ~7 nm. In order to fully take the advantage of the gradient, it is important to characterize devices with thicknesses of ≤ 7 nm. In the main text, they only presented devices having PtSe2 thicker than 10 nm. The observation of a different MR shape for the thickness of 8 nm itself suggests that the work should be extended to 2.5 nm < t < 7 nm regime (<2.5 nm was reported to be semiconducting).

Our Response:
We appreciate the reviewer's comment and suggestion. We agree that the characterization of a thin plasma-treated PtSe2 device (less than 7 nm) gives us further understanding about the relation between a defect-gradient and its impact on the Rashba effect. Thus, we fabricated a plasma-treated PtSe2 film of 6 nm through the same method and device structure as described in the main text. Then, we characterized its properties through magnetoelectrical measurements as described below.
We first performed the basic electrical characterization of a plasma-treated PtSe2 of 6 nm device. A temperature-dependent resistance result in Fig. R1a shows a semiconducting behavior as temperature increases, a measured resistance decreases. It is a different from a metallic behavior in plasma-treated PtSe2 films thicker than 10 nm (in the main text). Figure R1b supports the transition showing a Schottky contact at low temperature, in contrast to the Ohmic contact in all plasma-treated PtSe2 devices thicker than 10 nm in all temperature regions (see our response to comments #3). These results indicate the transition of a plasma-treated PtSe2 from metal to semiconductor as the thickness of PtSe2 decreases near 6 nm (Fig. R1c). Note that the metal to semiconductor transition in a pristine PtSe2 film near 8 nm was also reported [Adv. Mater. 29, 1604230 (2017); Nat. Commun. 9, 919 (2018)].

Fig. R1
Electrical characterization of a plasma-treated 6 nm PtSe2 film. a RRR of the device. b I-V characterization on various temperatures. c Temperature-dependent resistance of plasma-treated PtSe2 films from 6 nm to 15 nm.
Then, we analyzed a nonreciprocal MR feature in the same device, a plasma-treated PtSe2 of 6 nm. Figure  R2a,b displays the results by sweeping a magnetic field and an angle, respectively. Here, we used the same measurement condition (i.e. the direction x, y, and z) with the geometry in Fig. 4a (in the main text). Both results clearly show the nonreciprocal features in response to the polarity of a current and a magnetic field. Also, the analyzed magnetic field dependence for the nonreciprocal MR shows a linear behavior (the inset in Fig. R2a). These features follow well the nonreciprocal relation ΔR/R0 ~ (B × z) • . Compared to the plasma-treated PtSe2 films thicker than 8 nm, the nonreciprocal MR (ΔMR) of this 6 nm device showed two orders of magnitude lower value. Following the nonreciprocal relation, the ΔMR values in Fig. R2c show a linear behavior on applying currents. Further measurement with angle-dependent MR (ADMR) in  with the opposite direction of currents under B = +8 T at 10 K. c Current-dependent ΔMR at 10 K. d ADMR measured at various temperatures from 4 K to 300 K.
In response to the reviewer's comments and suggestion, we added the nonreciprocal MR results of a 6 nm plasma-treated PtSe2 ( Supplementary Information Fig. 14) and revised the manuscript in the main text (page 9).

#2. Reviewer's Comments:
It is not clear if the devices labelled as w/ plasma and w/o plasma in Fig.3 are the same devices. If yes, the presented results in Fig.3 would be expected since the thickness reduction due to plasma treatment would affect the thinnest material most as a result of PtSe2's unique thickness-dependent transport properties (A. Ciarrocchi et al., Nat. Commun (2018): device resistance drops more rapidly as thickness is reduced). Can they provide more information about their experimental methodology? E.g. what were the initial thicknesses of each crystal shown in Fig.3, before plasma treatment?

Our Response:
We appreciate the reviewer for pointing out the important labeling related to our experiment. The devices labelled as w/ plasma and w/o plasma in Fig. 3 were different devices. The thickness written in the text indicates the final thickness for a plasma-treated PtSe2 and for a PtSe2 without plasma treatment, which is mandatory because the resistivity of PtSe2 significantly varies with thickness as the reviewer mentioned.
First, for "t = 10 nm w/o plasma", we exfoliated a 10 nm PtSe2 flake on PDMS and transferred on an -Al2O3 substrate. Then, we made electrode contacts on top of the flake. Here we did not use any plasma treatment and there was no defect-gradient in the PtSe2 film.
Second, for "t = 10 nm w/ plasma", we exfoliated PtSe2 flakes thicker than 20 nm on PDMS and transferred on an -Al2O3 substrate. After that, we etched the flake down to 10 nm. The target thicknesses of films were controlled by plasma time, and the typical etching rate was 0.03 nm/s. Then, electrodes were made on top of the flake. Here, the initial thickness for a plasma-treated PtSe2 flake should be at least 7 nm-thicker than a target thickness to make a 7 nm defect-gradient in a PtSe2 film. For example, the initial thicknesses of "t = 10 nm w/ plasma" for SO-11 and SO-12 samples were 27 nm and 19 nm, respectively (see the bottom updated Table R1 for every sample w/ plasma). Even though we used several PtSe2 flakes having different initial thicknesses (before plasma treatment) to make the same target thickness, the fabricated samples showed almost the same behavior and thickness tendency in RRR and nonreciprocal MR.

Table R1
Summary of magneto-transport results measured in plasma-treated PtSe2 samples and the thickness information before and after plasma treatment.
As the reviewer mentioned, thickness-dependent transport properties of PtSe2 is unique, and we would like to note that our plasma-treated PtSe2 films also exhibited this thickness tendency (i.e. the resistivity of PtSe2 increases with decreasing thickness shown in Fig. R1c). In particular, we confirmed that the transition to a semiconducting behavior in a plasma-treated 6 nm PtSe2 film (Fig. R1a,c). Our plasmatreated PtSe2 films also exhibited the same thickness dependence as observed in pristine PtSe2 films [Adv. Mater. 29, 1604230 (2017); Nat. Commun. 9, 919 (2018)].
In response to the reviewer's comments, we added the detailed thickness information of the studied PtSe2 films (Supplementary Table 1), resistance results ( Supplementary Fig. 18) and further explanation about the device labelled w/ plasma and w/o plasma (page 7 and page 12) in the revised text.

#3. Reviewer's Comments:
If the authors are performing the plasma treatment before fabricating contacts, their contact/PtSe2 interface will not be the same as its pristine counterpart. Therefore, contact interface related effects could alter its transport properties. For the completeness of their work, I recommend authors perform similar directional magneto-transport measurements in a pristine device, followed by repeating the same measurements on it after creating the defect gradient.

Our Response:
We appreciate the reviewer's comment and suggestion for the contact between a PtSe2 flake and Au electrodes. Our nonreciprocal MR comes from charge transport through a PtSe2 longitudinal channel, located in the middle of the source and drain contact, which responses to an applied current and magnetic field. Then, the contact between a PtSe2 film and Au electrode could contribute to the response to a current and a magnetic field during charge injection/detection through their interface. Measured current-voltage (I-V) curves in our devices showed the Ohmic contact behavior in both a pristine PtSe2 device (w/o plasma) and a plasma-treated PtSe2 device (w/ plasma), shown in Fig. R3, which assured us reciprocal charge injection/detection that did not provide an external nonreciprocal factor to the MR measurement.

Fig. R3 I-V characteristics of the 10 nm PtSe2 films of a pristine (w/o plasma) and plasma-treated PtSe2 (w/ plasma).
The measurement the reviewer suggested is an ideal method to exclude any unexpected contact issues. However, if we first measure a pristine 10 nm PtSe2 device and perform plasma treatment to the pristine PtSe2 film to make a defect-gradient, the PtSe2 film will be etched and the thickness will decrease. Thus, we cannot compare a pristine 10 nm PtSe2 film with a plasma-treated 10 nm PtSe2 film in a single device. The other method could be that we first exfoliate a 20 nm pristine PtSe2 film and make electrodes on top of it, which has a contact of pristine PtSe2/Au electrode. Then, we etch it to make a 10 nm plasma-treated PtSe2 channel. However, the transport channel between the source and drain electrode will be composed of two different PtSe2 regions (a pristine PtSe2 of 20 nm under Au electrodes and a plasma-treated PtSe2 channel of 10 nm without top Au electrodes), which will definitely affect to overall charge transport.
In response to the reviewer's comments, we added the I-V characteristics of both pristine and plasmatreated 10 nm PtSe2 films ( Supplementary Fig. 11) and further explanation (page 9) in the revised manuscript.

#4. Reviewer's Comments:
Observation of a nonreciprocal MR up to RT is indeed interesting. It will be great if they could fully characterize a device at room temperature, as a function of applied current and magnetic field (directiondependent).

Our Response:
We greatly appreciate the reviewer's suggestion about room temperature measurement. As the reviewer mentioned, a nonreciprocal MR at room temperature is an attractive feature in particular to extend our approach to practical device application. Thus, we fabricated a plasma-treated PtSe2 film of 10 nm in the same method and device structure as described in the main text. Then, nonreciprocal MR at 300 K in the fabricated device was measured focusing on angle-dependence and magnetic field-dependence.
First, we measured ADMR at 300 K in the yx plane. Figure R4a-c shows ADMR results depending on a applied current. There was a clear inversion of MR according to the current polarity with I = +500 A and -500 A (Fig. R4a). The magnitude of the ADMR changed as a current increased from +100 A to +500 A (Fig. R4b). These asymmetric MR values (ΔMR), MR difference between the +y direction (and the y direction (), are plotted in Fig. R4c. Switching the polarity of a magnetic field also induced an inversion of a nonreciprocal MR (Fig. R4d). All results obtained for a plasma-treated 10 nm PtSe2 at 300 K followed well the nonreciprocal MR relation, ΔR/R0 ~ (B × z) • .
We further measured a magnetic field-dependent nonreciprocal MR at 300 K. Figure R4e shows different MR curves depending on a current of +500 A and 500 A, and their different MR values represent a nonreciprocal MR feature, as shown in Fig. R4f. This MR plot also shows linear dependence following the nonreciprocal relation.
Therefore, our plasma-treated PtSe2 system exhibited a nonreciprocal MR at 300 K, following well the relation of ΔR/R0 ~ (B × z) • . In addition, as we compared the ΔMR values in our new 10 nm plasmatreated PtSe2 device with the previous 10 nm plasma-treated PtSe2 device ( Fig. 4d and Supplementary Fig.  13), the observed MR and values were nearly identical as 0.01 % and 0.015, respectively, which indicated the reproducibility of our methodology to make a defect-gradient and a resulting nonreciprocal MR. In response to the reviewer's comments, we added nonreciprocal MR data at 300 K (Supplementary Fig. 13) and further explanation (page 9) in the revised manuscript.
#5. Reviewer's Comments: I think the present title is also a bit misleading. van der Waals nature of PtSe2 is not used in this work. I recommend the authors change the title to: Se-defect-gradient-induced Rashba effect in PtSe2 crystals. Just a suggestion.

Our Response:
We grateful for the reviewer's suggestion for the title. Our goal in this study is to build a layer-by-layer defect-gradient in a thin film system. In this respect, two dimensional van der Waals PtSe2 is the most proper material because of its binary components and the weak interlayer coupling to make composition difference along the layers. At first, if a target material is composed of elements with strong interlayer bonding like metals or oxides, it cannot sufficiently make a gradient through excessive energy dispersion by surface plasma treatment. If a target material consists of only reactive components (easy to etch), such as graphene and MoS2, it would be totally etched and could not make a gradient in spite of van der Waals films. Thus, the van der Waals nature of interlayer coupling along with constituents of noble atoms (e.g. Au, Pt, and Ag) is essential ingredients to develop a plasma-induced defect-gradient (i.e. an uniform gradient through a wide range along depth). This is why we would like to use the word "van der Waals" PtSe2 layers in the title.

#6. Reviewer's Comments:
What is the origin of the MR signal near B = 0 T (Fig 4-b)? It seems that part of the signal disappeared in thicker crystals and became stronger in the thinner, 8 nm sample. Could it be related to the MR signals reported in Pt defect dominated PtSe2 devices (Ref 15)?

Our Response:
We appreciate the reviewer's comments about the important issue on low-field MR. As the reviewer mentioned, a small MR signal near B = 0 T is related to a Pt defect-induced magnetism as reported [Nat. Nanotech. 14, 674 (2019); Nat. Commun. 11, 4806 (2020)]. We used a PtSe2 crystal purchased from HQ graphene that was also the source material for the study of the above two previous reports. The synthetic methods for PtSe2 crystal from HQ graphene is known to induce slight Pt-vacancy defects which lead to long-range magnetic ordering at low temperature. The following Fig. R5 represents anisotropic magnetoresistance (AMR) signals in our PtSe2 devices. The AMR signal clearly exhibited kink-like features near the coercive field for forward and backward magnetic field sweeps (the inset in Fig. R5a), which is conventional characteristics of a ferromagnet. We observed this AMR signal for both the PtSe2 films without (Fig. R5a-c) and with plasma treatment (Fig. R5d-f). This implies that our Se defect-gradient and induced nonreciprocal transport features are not related with Pt defect-induced AMR. In addition, as the thickness of a PtSe2 film increased, the AMR signal got smaller and eventually disappeared for the film with thickness over 15 nm, which was consistent with the previous reports. In response to the reviewer's comments, we added the AMR analysis ( Supplementary Fig. 9) and further explanation about AMR (page 8) in the revised manuscript.
In summary, the major changes we made in response to the reviewer's comments are as follows.  Supplementary Fig. 9 for AMR features in PtSe2 films. 4. Addition of Supplementary Fig. 11 for the Ohmic contact behavior between a PtSe2 film and an Au electrode before and after plasma treatment. 5. Addition of Supplementary Fig. 13 for nonreciprocal MR features at room temperature. 6. Addition of Supplementary Fig. 14 for nonreciprocal MR features of a plasma-treated PtSe2 of 6 nm. 7. Addition of Supplementary Fig. 18 for temperature-dependent resistance depending on thickness and plasma treatment. 8. Addition of thickness information before and after plasma treatment in Supplementary Table 1 and further explanation for device fabrication in the main text. 9. Addition of Supplementary Fig. 20 for DFT band structures of the pristine PtSe2 film.

Reviewer #2 (Remarks to the Author):
Controlling defects in crystals is an effective technique for manipulating the physical properties and functionalities. In this manuscript, authors demonstrate that is can be also useful for symmetry engineering, providing a new direction of defect engineering. They report that layer-by-layer defect gradients in two dimensional PtSe2 give rise to the spatial inversion symmetry and resultant Rashba effect, which was sensitively probed by nonreciprocal transport measurement, an intrinsic rectification effect reflecting the broken inversion symmetry. They also found that nonreciprocal transport survives up to room temperature, which is scientifically interesting and important for the application.
I feel that results are great and worth publishing in Nature Communications. However, I think the authors should further consider and clarify the following point before accepting the manuscript.

Our Response:
We greatly appreciate the reviewer's assessment on the novelty and importance of our results. As the reviewer mentioned, defect engineering is an effective technique to manipulate the material's properties which have been used for long time in the semiconductor industry, and it has extended its utility to the nanotechnology and, in particular, the two-dimensional materials. We believe that our approach to utilize defects as a layer-by-layer gradient would be a novel method to induce inversion symmetry breaking and use for practical device application. The following is detailed discussions and corrections in response to the reviewer's comments to improve the quality of this manuscript.

#1. Reviewer's Comments:
I would like to know the effect of carrier number change by defect engineering. In addition to the symmetry change, carrier number change cannot be avoided, which affects the magnitude of the nonreciprocal transport (Nat. Phys. 13, 578 (2017), Nat. Commun. 10, 4510 (2019)).

Our Response:
We appreciate the reviewer's comment on the carrier concentration by defect engineering. We performed Hall measurements to check the carrier concentration for a pristine 10 nm PtSe2 film (w/o plasma) and a plasma-treated 10 nm PtSe2 film (w/ plasma) in comparison. The carrier concentration for the PtSe2 w/o plasma was 3.20ⅹ10 21 cm -3 at 2 K and 1.01ⅹ10 22 cm -3 at 300 K. In contrast, plasma treatment reduced the concentration as 3.6ⅹ10 20 cm -3 at 2 K and 1.47ⅹ10 21 cm -3 at 300 K. Figure R6 shows the temperaturedependent carrier concentration and mobility for the PtSe2 films of 10 nm w/ and w/o plasma treatment. In response to the reviewer's comments, we added the information about carrier concentration and mobility ( Supplementary Fig. 7) and further explanation for the carrier concentration (page 7) in the revised manuscript.

#2. Reviewer's Comments:
What is the carrier number and expected Fermi level (EF) in each samples?

Our Response:
Following up the above response, we calculated the carrier concentration and Fermi level of 10 nm PtSe2 films w/ and w/o plasma treatment. In a pristine PtSe2 film of 10 nm, the Fermi level was estimated as 0.791 eV at 2 K with the carrier concentration of 3.20ⅹ10 21 cm -3 . In contrast, the estimated Fermi level of a plasma-treated PtSe2 of 10 nm is 0.185 eV with the carrier concentration of 3.60ⅹ10 20 cm -3 at 2 K, respectively. It was reported that low carrier concentration and Fermi level were favorable to induce higher magnitude of the nonreciprocal magnetoresistance in the Rashba systems [Nat. Phys. 13, 578 (2017); Nat. Commun. 10, 4510 (2019)]. Compared to these previous reports, our plasma-treated PtSe2 system has rather higher carrier concentration which could be attributed to the low nonreciprocal coefficient () and nonreciprocal MR.
#3. Reviewer's Comments: 3. I am interested in whether spin-orbit coupling (SOC) other than Rashba effect can emerge in this system. What is the (effective) point group of this system? Are there any Zeeman-type SOC in such as 2H-type transition metal dichalcogenides? (For example, out-of-plane components of SOC can be probed by applying the magnetic field perpendicular to the two-dimensional materials. (Sci. Adv. 3, e1602390 (2017), Phys. Rev. Lett. 120, 266802 (2018)))

Our Response:
We grateful for the reviewer's comments on the structure of PtSe2. The pristine PtSe2 was 1T-phase possessing the D3d point group composed of three atomic layers with Pt and Se atoms (Fig. R7). Importantly, this pristine 1T-phase PtSe2 preserves inversion symmetry, which doesn't show any spin splitting in its band structure (please see Supplementary Fig. 20). Our defect-gradient PtSe2 thin films would be away from the inversion-symmetric D3d point group, and once we consider the layer-by-layer variation of the defect concentration, it is effectively close to the C3v point group symmetry. The Zeemantype SOC occurred in 2H-type TMDC monolayer appears through the combination between the absence of inversion symmetry and the presence of horizontal mirror symmetry; the spin splitting induced by inversion asymmetry shows the out-of-plane spin texture that is protected by horizontal mirror symmetry. In our defect-gradient PtSe2 thin films, the horizontal mirror symmetry as well as the inversion symmetry is severely broken, (mostly) supporting the Rashba-type SOC and helical spin texture. Furthermore, obviously, we could not detect nonreciprocal MR behavior in our magneto-transport measurement under the perpendicular magnetic field Bz (see Fig. 3d-f in the main text) regardless of plasma treatment. Thus, we can infer that the observed nonreciprocal behavior in plasma-treated PtSe2 film comes from the Rashba effect induced by a defect-gradient. In response to the reviewer's comments, we added the band structures of pristine PtSe2 film ( Supplementary Fig. 20) and further explanation (page 11) in the revised manuscript.

Reviewer's Comments:
There is a kink structure in magnetoresistance (Fig. 4 b, Fig. S13). What is the possible origin of this strange behavior?

Our Response:
We appreciate the reviewer's comments about the important issue on low-field MR. As the reviewer mentioned, a small MR signal near B = 0 T is related to a Pt defect-induced magnetism as reported [Nat. Nanotech. 14, 674 (2019); Nat. Commun. 11, 4806 (2020)]. We used a PtSe2 crystal purchased from HQ graphene that was the same material with the study of the above two previous reports. The synthetic methods for PtSe2 crystal from HQ graphene is known to induce slight Pt-vacancy defects, which lead to long-range magnetic ordering at low temperature. The following Fig. R8 represents anisotropic magnetoresistance (AMR) signals in our PtSe2 devices. The AMR signal clearly exhibited kink-like features near the coercive field for forward and backward magnetic field sweeps (the inset in Fig. R8a), which is conventional characteristics of a ferromagnet. We observed this AMR signal for both the PtSe2 films without (Fig. R8a-c) and with plasma treatment (Fig. R8d-f). This implies that our Se defect-gradient and induced nonreciprocal transport features are not related with Pt defect-induced AMR. In addition, as the thickness of a PtSe2 film increased, the AMR signal got smaller and eventually disappeared for the thicker film over 15 nm, which was consistent with the previous reports. In response to the reviewer's comments, we added the AMR results ( Supplementary Fig. 9) and further explanation (page 8) in the revised manuscript.

Reviewer's Comments:
In 8 nm thick film, magneto resistance (MR) is negative, while the other samples show the positive MR. in 8 nm film. What is the difference?

Our Response:
We appreciate the reviewer's comment about a negative MR under By in a 8 nm PtSe2 film. There are several factors which can induce a negative MR in thin film systems such as weak localization, Kondo effect, chiral anomaly, magnetism, and current jetting. relation which represented the proportional relation between the  and B 2 (Fig. R9b). In addition, the MR under By also displayed a negative MR (Supplementary Fig. 15c and Fig. R9c). However, as this negative MR under By and its origin have not been studied yet, it is difficult to conclude the relation between the negative MR under By and the chiral anomaly.
-Weak localization can result in a negative MR when a magnetic field is applied to the perpendicular direction (Bz). However, our sample showed the ordinary parabolic behavior of MR under Bz at 2 K and persisted up to Bz = 8 T that was far away from the low magnetic field region for weak localization usually less than 4 T. This absence of weak localization effect in our PtSe2 film can exclude its effect on the negative MR under By.
-Magnetic ordering can induce a negative MR, generally we called anisotropic magnetoresistance (AMR) in a ferromagnet. Both 8 nm and 10 nm PtSe2 films clearly showed small AMR signals, i. e. the hysteresis feature during the forward and backward magnetic field sweep (Fig. R8a,b). It was reported that this AMR came from Pt defect-induced magnetism in PtSe2 [Nat. Nanotech. 14, 674 (2019); Nat. Commun. 11, 4806 (2020)]. Thus, the negative MR shown exclusively for a 8 nm PtSe2 film can be distinguished from the AMR (Fig. R8d and Fig. R9c), and we can exclude the effect from magnetic ordering.
-Current jetting can be induced by non-homogenous current distribution. It usually occurs in a large sample which does not have a well defined channel and electrodes [Nat. Commun. 7, 11615 (2016)]. Thus, it is not related to our device with a few micron-meter scale electrode patterned by photolithography.
-Kondo effect can be a source for a negative MR in a thin film system. As the Kondo effect comes from the isotropic s-d exchange interaction between conduction electrons and localized magnetic moments, the MR resulting from the Kondo effect should be independent of the magnetic field direction [Adv. Mater. 33, 2005465 (2020)]. In our PtSe2 system, there is clear difference in MR curves under Bx (Fig. R9b) and By (Fig. R9c), even if we consider a nonreciprocal MR feature. Even though we have checked possible factors for the origin of a negative MR under By in a 8 nm PtSe2 film, we could not clearly conclude the origin. This negative MR under By exhibited not only in a plasma-treated 8 nm PtSe2 film but also in a pristine 8 nm PtSe2 film (Fig. R10), which might indicate that a defect-gradient would not play an critical role for a negative MR under By. We note that all PtSe2 films (w/ and w/o plasma) thicker than 10 nm showed a positive MR under By (and Bx as well). x. c Magnetic field (By)-dependent conductivity.

Our Response:
As mentioned by the referee, the sub-band structure of the defect-gradient PtSe2 thin films could play a crucial role in the non-reciprocal transport. According to our DFT calculation in Fig 5c, we have two different types of sub-bands depending on their spatial distribution. The one sub-bands (colored by dark brown) mostly located in the defect-gradient region feel the strong inversion-breaking field, resulting in the large Rashba-type spin splitting. The other sub-bands (colored by white) located in the pristine region hardly see the inversion-asymmetry, showing no spin splitting.
In summary, the major changes we made in response to the reviewer's comments are as follows.
1. Change of the Fig. 1b image and detailed explanation for a device geometry in the main text. 2. Addition of Supplementary Fig. 7 for carrier concentration and Fermi level change after plasma treatment. 3. Addition of Supplementary Fig. 9 for AMR features in PtSe2 films. 4. Addition of Supplementary Fig. 11 for the Ohmic contact behavior between a PtSe2 film and an Au electrode before and after plasma treatment. 5. Addition of Supplementary Fig. 13 for nonreciprocal MR features at room temperature. 6. Addition of Supplementary Fig. 14 for nonreciprocal MR features of a plasma-treated PtSe2 of 6 nm. 7. Addition of Supplementary Fig. 18 for temperature-dependent resistance depending on thickness and plasma treatment. 8. Addition of thickness information before and after plasma treatment in Supplementary Table 1 and further explanation for device fabrication in the main text. 9. Addition of Supplementary Fig. 20 for DFT band structures of the pristine PtSe2 film.

Reviewer's Comments:
This manuscript presents an investigation of the potential for engineering a Rashba field within a 2D material using a vertical gradient in the density of structural defects. The material of choice is PtSe2, and the authors verify the creation of the structural defects with a combination of electron microscopy and spectroscopy. The impact of the potential Rashba field is measured using longitudinal magneto-transport, and the authors demonstrate the onset of an asymmetry in the MR as a function of current direction in samples with defect gradients that is not present in pristine samples. A theoretical basis for this effect is presented using a combination of referencing to prior literature and the presentation of DFT calculations of pristine and defected material.
The material characterization is extremely thorough and the authors demonstrate the creation of the defect gradient that they are attempting to engineer in a very compelling fashion. Similarly, the presence of the asymmetry in magnetoresistance is also clear and indexing to control samples provides a compelling argument that it is related to the defect incorporation. The theoretical foundations for assigning this MR to a Rashba field was less clear to me, and I am unsure if this is simply an issue of clarity in the manuscript or if there are more fundamental issues. As a result, while I believe there is the potential for impactful science here I would like to see a revised manuscript that addresses the following points before advocating for publication.

Our Response:
We greatly appreciate the reviewer's meticulous assessment and comments for recognizing the novelty and importance of our results. We believe that a defect-gradient is a novel approach to develop a layerby-layer structure to induce the inversion symmetry breaking and Rashba field. Therefore, we introduced TEM analysis, magnetotransport measurement, and DFT calculation to prove our defect-gradient structure and propose its utility for practical device application and scientific research. The following is detailed discussions and corrections in response to the reviewer's comments to improve the quality of this manuscript.

#1. Reviewer's Comments:
There is some confusion regarding the orientation of the respective fields. First, it is my understanding that the native spin orbit coupling (SOC) in PtSe2 is out of plane (in the z direction according to the authors convention). Since this is the same direction as the Rashba field proposed as arising from the defects I would appreciate some discussion of how these two effects are distinct and/or can be disentangled. Second, the labeling in the SEM image of their device in Fig 1 is confusing. The image is zoomed in so that only a series of stripes is visible, which I take to be the transport channels of the PtSe2. Under this convention the authors appear to be proposing a transverse current while their measurements are longitudinal. I think a much more careful description of the sample geometry and the symmetry of the relevant effects is warranted.

Our Response:
We appreciate the reviewer's comments on the structure of PtSe2. We used 1T-phase PtSe2 in this study, possessing the D3d point group composed of three atomic layers with Pt and Se atoms. This implies that the pristine 1T-phase PtSe2 thin films preserve inversion symmetry and show the spin degenerate band structure. In other words, there is no spin splitting in its band structure. (see Supplementary Fig. 20.) Our defect-gradient PtSe2 system would be different from the symmetric D3d point group of the pristine, and it is effectively close to the C3v point group, supporting the effective inversion-breaking electric field perpendicular to the plane. As a result of this out-of-plane inversion-breaking field, the defect-gradient PtSe2 thin films can possess the Rashba-type SOC and helical spin texture. The disentanglement of Zeeman type and Rashba type spin-splitting can be probed by applying the magnetic field perpendicular to the two-dimensional materials [Sci. Adv. 3, e1602390 (2017); Phys. Rev. Lett. 120, 266802 (2018)]. However, we could not detect any nonreciprocal MR behavior in our magneto-transport measurement under the out-of-plane magnetic field (Fig. 3d-f in the main text). Thus, we can infer that the observed nonreciprocal behavior originated from the Rashba effect due to a defect-gradient in a PtSe2 film.
We agree the reviewer's comment on the SEM image. To make it clear, we revised colors, marks, and explanation in Fig. 1b. Here, we would like to mention that we used a large size of Au electrodes because of our plasma treatment process. For the plasma treatment, we used an -Al2O3 substrate that was stable for the etching with a SF6 gas, in contrast to a conventional Si/SiO2 substrate that was harshly etched (20 times higher etching rate than PtSe2). Unfortunately, this -Al2O3 substrate could not be used for our electron beam lithography system. Thus, we used photolithography to make electrode patterns, which made relatively the large width of 4-terminal electrodes and the rough edge of patterns.
In response to the reviewer's comments, we modified Fig. 1b and added Supplementary Fig. 20 with further explanation (page 5, page 8, page 11, and page 19) in the revised manuscript.

#2. Reviewer's Comments:
The role of the DFT in validating the proposed mechanism is not clear to me. The two plots presented are DFT with and without spin orbit coupling, but as noted above PtSe2 has native SOC that must be taken into account. Are the authors explicitly taking into account different levels of defect incorporation into account in their calculations? If so, there is a high computational cost for the more heavily defected layers and I would be interested to hear more about how they manage this calculation. If not, then the applicability of this DFT work is not clear. How does it support the hypothesis that the defect gradient is giving rise to a spin-orbit field with the appropriate symmetry to generate the experimental results? If the authors are just manually tuning the strength of SOC to mimic what they expect to happen with the different defect densities then this needs to be clearly stated and the implications of this approach need to be more thoroughly discussed.

Our Response:
Applying the first-principles DFT method to the system with defects requires some approximations for the treatment of the disordered defects. A direct approach is to use a supercell approximation to consider the isolated defect. As the reviewer pointed out, such calculations are computationally very demanding. To circumvent this difficulty, we used virtual crystal approximation (VCA) that is a computationally less expensive approach and at the same time can capture the averaged effect of the disordered defects. In VCA, we construct our thin film geometry with the primitive periodicity using virtual atoms averaged by Se and vacancy, indeed producing the defect distribution shown in Fig. 2b. This technique has widely used in various band-structure calculations. Previous work has demonstrated good accuracy in some semiconductor and ferromagnetic materials (see R1-R7 below). In Ref. R7, in particular, supercell calculation and VCA showed consistent results. In addition to this, VCA has been successfully applied to studies on topological phase transitions that require very precise handling of structural symmetry and SOC [R8-R14]. It seems that VCA is sufficient to examine how the defect qualitatively affects the symmetry of the system and the strength of the SOC, especially in our work.
The Rashba splitting is a combined effect of SOC and inversion-breaking field. The purpose of our calculations is to reveal that Rashba-type spin-splitting occurs due to defect-gradient in our experiments. Therefore, we tried to show that both atomic SOC and defect-gradient are necessary for the finite spinsplitting. First, as shown in Fig. 5b,c, we can confirm that SOC is necessary to generate Rashba-type spinsplitting since we do not see any splitting without SOC in the defect-gradient system. On the other hand, the inversion-breaking induced by the defect-gradient has been also turned out to be an essential ingredient for the spin-splitting ( Supplementary Fig. 20). The splitting does not appear regardless of SOC in the inversion symmetric system of a pristine PtSe2 film.
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